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diff --git a/pages/measure-and-integration/bochner-integral/index.md b/pages/measure-and-integration/bochner-integral/index.md new file mode 100644 index 0000000..5517934 --- /dev/null +++ b/pages/measure-and-integration/bochner-integral/index.md @@ -0,0 +1,9 @@ +--- +title: Bochner Integral +parent: Measure and Integration +nav_order: 3 +has_children: true +has_toc: false +--- + +# {{ page.title }} diff --git a/pages/measure-and-integration/index.md b/pages/measure-and-integration/index.md new file mode 100644 index 0000000..841e941 --- /dev/null +++ b/pages/measure-and-integration/index.md @@ -0,0 +1,8 @@ +--- +title: Measure and Integration +nav_order: 2 +has_children: true +has_toc: false +--- + +# {{ page.title }} diff --git a/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md new file mode 100644 index 0000000..a77cf9a --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md @@ -0,0 +1,27 @@ +--- +title: Almost Everywhere +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 1 +--- + +# {{ page.title }} + +{% definition Almost Everywhere %} +We say that a property $P(x)$ depending on $x \in X$ +holds *almost everywhere* (abbreviated by *a.e.*) or for *almost all $x \in X$* if +the set of points where it does not hold has measure zero. +{% enddefinition %} + +In other words, $P(x)$ a.e. iff +$\mu(\set{x \in X : \neg P(x)}) = 0$. + +{% theorem %} +Let $f : X \to \overline{\RR}$ be a nonnegative measurable function. Then + +$$ +\int_X f \, d\mu = 0 +$$ + +holds if and only if $f$ vanishes almost everywhere. +{% endtheorem %} diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md new file mode 100644 index 0000000..67f0996 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md @@ -0,0 +1,77 @@ +--- +title: Convergence Theorems +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 2 +--- + +# {{ page.title }} + +For all statements on this page, +assume that $(X,\mathcal{A},\mu)$ is a measure space. + +{% theorem * Monotone Convergence Theorem %} +For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ be a measurable function. +If $0 \le f_n \le f_{n+1}$ almost everywhere, then + +$$ +\int_X \lim_{n \to \infty} f_n \, d\mu = \lim_{n \to \infty} \int_X f_n \, d\mu. +$$ +{% endtheorem %} + +Note that the pointwise limit $\lim_{n \to \infty} f_n$ always exists and is measurable by this proposition. + +{% lemma * Fatou’s Lemma %} +For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ be a nonnegative measurable function. Then + +$$ +\int_X \liminf_{n \to \infty} f_n \, d\mu \le \liminf_{n \to \infty} \int_X f_n \, d\mu. +$$ +{% endlemma %} + +In the following proof we omit $X$ and $d\mu$ for visual clarity. + +{% proof %} +By definition, we have $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} g_n$, where $g_n = \inf_{k \ge n} f_k$. +Now $(g_n)$ is a monotonic sequence of nonnegative measurable functions. +By the +[Monotone Convergence Theorem](#monotone-convergence-theorem) + +$$ +\int \liminf_{n \to \infty} f_n = \lim_{n \to \infty} \int g_n. +$$ + +For all $k \ge n$ one has $g_n \le f_k$, hence +$\int g_n \le \int f_k$ by the monotonicity of the integral. +This implies + +$$ +\int g_n \le \inf_{k \ge n} \int f_k +$$ + +for all $n \in \NN$. In the limit $n \to \infty$ we obtain + +$$ +\lim_{n \to \infty} \int g_n +\le \liminf_{n \to \infty} \int f_n +$$ + +thereby completing the proof. +{% endproof %} + +{% theorem * Dominated Convergence Theorem %} +Let $(X,\mathcal{A},\mu)$ be a measure space. +For each $n \in \NN$ let $f_n : X \to \overline{\RR}$ (or $\CC$) be a measurable function. +Suppose that the pointwise limit $f = \lim_{n \to \infty} f_n$ exists almost everywhere. +Suppose further that there exists an integrable function $g : X \to \overline{\RR}$ +such that $\abs{f_n} \le g$ almost everywhere for all $n \in \NN$. +Then the functions $f_n$ and $f$ are all integrable, and + +$$ +\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu. +$$ +{% endtheorem %} + +{% proof %} +TODO +{% endproof %} diff --git a/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md new file mode 100644 index 0000000..6e5179c --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md @@ -0,0 +1,14 @@ +--- +title: Fubini Theorem +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 2 +--- + +# {{ page.title }} + +{% theorem %} +{% endtheorem %} + +{% proof %} +{% endproof %} diff --git a/pages/measure-and-integration/lebesgue-integral/index.md b/pages/measure-and-integration/lebesgue-integral/index.md new file mode 100644 index 0000000..a857d95 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/index.md @@ -0,0 +1,112 @@ +--- +title: Lebesgue Integral +parent: Measure and Integration +nav_order: 2 +has_children: true +has_toc: false +--- + +# {{ page.title }} + +For this entire section we fix a measure space $(X,\mathcal{A},\mu)$. + +## Integration of Nonnegative Step Functions + +{% definition %} +Let $f : X \to \RR$ be a nonnegative step function +with representation $f = \sum_{i=1}^n \alpha_i \chi_{A_i}$, +where $\alpha_1, \ldots, \alpha_n \ge 0$ and +$A_1, \ldots, A_n \in \mathcal{A}$. +We define the *integral of $f$ on $X$ with respect to $\mu$* by + +$$ +\int_X f \, d\mu = \sum_{i=1}^n \alpha_i \, \mu(A_i) \in [0,\infty]. +$$ +{% enddefinition %} + +TODO: This does not depend on the representation of $f$. + +## Integration of Nonnegative Measurable Functions + +{% theorem Approximation by Step Functions %} +Every nonnegative measurable function $f : X \to \overline{\RR}$ +is the pointwise limit of an increasing sequence $(s_n)$ of +nonnegative step functions $s_n : X \to \RR$. +{% endtheorem %} + +{% definition %} +Let $f : X \to \overline{\RR}$ be a nonnegative measurable function +and let $(s_n)$ be a sequence of nonnegative step functions +with $s_n \uparrow f$. +We define the *integral of $f$ on $X$ with respect to $\mu$* by + +$$ +\int_X f \, d\mu = \lim_{n \to \infty} \int_X s_n \, d\mu \in [0,\infty]. +$$ +{% enddefinition %} + +## Integrable Functions + +Recall that the positive and (flipped) negative parts +of a function $f : X \to \overline{R}$ are defined by + +$$ +f^+ = \max(f,0) \qquad +f^- = \max(-f,0), +$$ + +and that $f$ is measurable if and only if both $f^+$ and $f^-$ are measurable. +We have $f = f^+\! - f^-$. + +{% definition Integrable Function, Lebesgue Integral %} +A measurable function $f : X \to \overline{\RR}$ is said to be +*integrable on $X$ with respect to $\mu$* if the integrals + +$$ +\int_X f^+ \, d\mu, \qquad \int_X f^- \, d\mu +\tag{$*$} +$$ + +are both finite. +In this case the *(Lebesgue) integral of $f$ on $X$ with respect to $\mu$* is defined as + +$$ +\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \RR. +$$ +{% enddefinition %} + +Sometimes it is convenient to have a slightly more general notion of integrability: + +{% definition Quasi-Integrable Function %} +A measurable function $f : X \to \overline{\RR}$ is said to be +*quasi-integrable on $X$ with respect to $\mu$* if at least one of the integrals +$(*)$ is finite. +In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as + +$$ +\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu \in \overline{\RR}. +$$ +{% enddefinition %} + +{% definition %} +A measurable function $f : X \to \CC$ is said to be +*integrable on $X$ with respect to $\mu$* if +$\Re f$ and $\Im f$ are integrable on $X$ with respect to $\mu$. +In this case the *integral of $f$ on $X$ with respect to $\mu$* is defined as + +$$ +\int_X f \, d\mu = \int_X \Re f \, d\mu + i \int_X \Im f \, d\mu \in \CC. +$$ +{% enddefinition %} + +## Integration on Measurable Subsets + +{% definition %} +For any measurable subset $A \subset X$ we define +the *integral on $A$* of a (quasi-)integrable function $f : X \to \overline{\RR}$ (or $\CC$) by + +$$ +\int_A f \, d\mu = +\int_X \chi_A f \, d\mu. +$$ +{% enddefinition %} diff --git a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md new file mode 100644 index 0000000..023c253 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md @@ -0,0 +1,36 @@ +--- +title: The L<sup>p</sup> Spaces +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 4 +--- + +# {{ page.title }} + +{% definition %} +Let $(X,\mathcal{A},\mu)$ be a measure space and let $p \in [1,\infty)$. +We write $\mathscr{L}^p(X,\mathcal{A},\mu)$ for the set of all +measurable functions $f : X \to \KK$ such that $\abs{f}^p$ is integrable. +For such $f$ we write + +$$ +\norm{f}_p = {\bigg\lparen\int_X \abs{f}^p \, d\mu\bigg\rparen}^{\!1/p}. +$$ +{% enddefinition %} + +{% proposition %} +Endowed with pointwise addition and scalar multiplication +$\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space. +{% endproposition %} + +{% proposition %} +$\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$. +{% endproposition %} + +{% theorem * Young Inequality %} +Consider $p,q > 1$ such that $1/p + 1/q = 1$. Then + +$$ +a \cdot b \le \frac{a^p}{p} + \frac{b^q}{q} \qquad \forall a,b \ge 0. +$$ +{% endtheorem %} diff --git a/pages/measure-and-integration/lebesgue-integral/transformation-formula.md b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md new file mode 100644 index 0000000..6f02bc8 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md @@ -0,0 +1,14 @@ +--- +title: Transformation Formula +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 3 +--- + +# {{ page.title }} + +{% theorem %} +{% endtheorem %} + +{% proof %} +{% endproof %} diff --git a/pages/measure-and-integration/measure-theory/borels-sets.md b/pages/measure-and-integration/measure-theory/borels-sets.md new file mode 100644 index 0000000..737a7c8 --- /dev/null +++ b/pages/measure-and-integration/measure-theory/borels-sets.md @@ -0,0 +1,33 @@ +--- +title: Borel Sets +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 2 +--- + +# {{ page.title }} + +{% definition Borel Sigma-Algebra, Borel Set %} +The *Borel σ-algebra* $\mathcal{B}(X)$ on a topological space $X$ is +the σ-algebra generated by its open sets. +The elements of $\mathcal{B}(X)$ are called *Borel(-measurable) sets*. +{% enddefinition %} + +That is, $\mathcal{B}(X) = \sigma(\mathcal{O})$, +where $\mathcal{O}$ is the collection of open sets in $X$. +It is also true that $\mathcal{B}(X) = \sigma(\mathcal{C})$, +where $\mathcal{C}$ is the collection of closed sets in $X$. + +{% definition Borel Function %} +If $(X,\mathcal{A})$ is a measure space +and $Y$ is a topological space, +then a function $f : X \to Y$ is called *measurable*, +or a *Borel function*, +if it is measurable with respect to $\mathcal{A}$ and +the Borel σ-algebra on $Y$. +{% enddefinition %} + +{% definition Borel Measure %} +A *Borel measure* on a topological space $X$ +is any measure on the Borel σ-algebra of $X$. +{% enddefinition %} diff --git a/pages/measure-and-integration/measure-theory/index.md b/pages/measure-and-integration/measure-theory/index.md new file mode 100644 index 0000000..575c945 --- /dev/null +++ b/pages/measure-and-integration/measure-theory/index.md @@ -0,0 +1,9 @@ +--- +title: Measure Theory +parent: Measure and Integration +nav_order: 1 +has_children: true +has_toc: false +--- + +# {{ page.title }} diff --git a/pages/measure-and-integration/measure-theory/measurable-maps.md b/pages/measure-and-integration/measure-theory/measurable-maps.md new file mode 100644 index 0000000..5b7a76e --- /dev/null +++ b/pages/measure-and-integration/measure-theory/measurable-maps.md @@ -0,0 +1,27 @@ +--- +title: Measurable Maps +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 3 +--- + +# {{ page.title }} + +{% definition Measurable Map %} +Suppose $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces. +We say that a map $f: X \to Y$ is *measurable* (with respect to $\mathcal{A}$ and $\mathcal{B}$) if +$f^{-1}(B) \in \mathcal{A}$ for all $B \in \mathcal{B}$. +{% enddefinition %} + +{% proposition %} +The composition of measurable maps is measurable. +{% endproposition %} + +It is sufficient to check measurability for a generator: + +{% proposition %} +Suppose that $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces, +and that $\mathcal{E}$ is a generator of $\mathcal{B}$. +Then a map $f : X \to Y$ is measurable iff +$f^{-1}(E) \in \mathcal{A}$ for every $E \in \mathcal{E}$. +{% endproposition %} diff --git a/pages/measure-and-integration/measure-theory/measures.md b/pages/measure-and-integration/measure-theory/measures.md new file mode 100644 index 0000000..637ab0c --- /dev/null +++ b/pages/measure-and-integration/measure-theory/measures.md @@ -0,0 +1,29 @@ +--- +title: Measures +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 4 +--- + +# {{ page.title }} + +{% definition %} +A *measure* on a σ-algebra $\mathcal{A}$ on a set $X$ +is mapping $\mu : \mathcal{A} \to [0,\infty]$ such that + +- $\mu(\varnothing) = 0$, +- for every sequence $(A_n)_{n \in \NN}$ of + pairwise disjoint sets $A_n \in \mathcal{A}$ + + $$ + \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen + = \sum_{n=0}^{\infty} \mu(A_n). + $$ +{% enddefinition %} + +{% definition Measure Space %} +A *measure space* is a triple $(X,\mathcal{A},\mu)$ of +a set $X$, +a σ-algebra $\mathcal{A}$ on $X$ +and a measure $\mu$ on $\mathcal{A}$. +{% enddefinition %} diff --git a/pages/measure-and-integration/measure-theory/sigma-algebras.md b/pages/measure-and-integration/measure-theory/sigma-algebras.md new file mode 100644 index 0000000..5d22f6b --- /dev/null +++ b/pages/measure-and-integration/measure-theory/sigma-algebras.md @@ -0,0 +1,50 @@ +--- +title: σ-Algebras +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 1 +--- + +# {{ page.title }} + +{% definition Sigma-Algebra, Measurable Space, Measurable Set %} +A *σ-algebra* on a set $X$ is a collection $\mathcal{A}$ of subsets of $X$ such that + +- $X$ belongs to $\mathcal{A}$, +- if $A \in \mathcal{A}$, then $X \setminus A \in \mathcal{A}$, +- the union of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$. + +A *measurable space* is a pair $(X,\mathcal{A})$ consisting of +a set $X$ and a σ-algebra $\mathcal{A}$ on $X$. \ +The subsets of $X$ belonging to $\mathcal{A}$ are called *measurable sets*. +{% enddefinition %} + +{% example %} +On every set $X$ we have the σ-algebras $\braces{\varnothing,X}$ and $\mathcal{P}(X)$. +{% endexample %} + +{% proposition %} +If $\mathcal{A}$ is *σ-algebra* on a set $X$, then: + +- $\varnothing$ belongs to $\mathcal{A}$, +- if $A,B \in \mathcal{A}$, then $B \setminus A \in \mathcal{A}$, +- the intersection of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$. +{% endproposition %} + +## Generated {{ page.title }} + +{% proposition Intersection of σ-Algebras %} +If $\braces{\mathcal{A}_i}$ is a family of σ-algebras on a set $X$, +then $\bigcap_i \mathcal{A}_i$ is a σ-algebra on $X$. +{% endproposition %} + +{% definition Generated σ-Algebras %} +Suppose $\mathcal{E}$ is any collection of subsets of a set $X$. +The *σ-algebra generated by $\mathcal{E}$*, denoted by $\sigma(\mathcal{E})$, is +defined to be the intersection of all σ-algebras on $X$ containing $\mathcal{A}$. +{% enddefinition %} + +By the previous proposition, $\sigma(\mathcal{E})$ is in fact a σ-algebra on $X$. + +## Products of {{ page.title }} + diff --git a/pages/measure-and-integration/measure-theory/signed-measures.md b/pages/measure-and-integration/measure-theory/signed-measures.md new file mode 100644 index 0000000..77b2416 --- /dev/null +++ b/pages/measure-and-integration/measure-theory/signed-measures.md @@ -0,0 +1,33 @@ +--- +title: Signed Measures +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 10 +--- + +# {{ page.title }} + +{% definition Signed Measure %} +A *signed measure* on a σ-algebra $\mathcal{A}$ on a set $X$ +is a mapping $\mu : \mathcal{A} \to [-\infty,\infty]$ such that +{: .mb-0 } + +- $\mu(\varnothing) = 0$, +- either there is no $A \in \mathcal{A}$ with $\mu(A) = -\infty$ + or there is no $A \in \mathcal{A}$ with $\mu(A) = \infty$, +- for every sequence $(A_n)_{n \in \NN}$ of + pairwise disjoint sets $A_n \in \mathcal{A}$ + {: .my-0 } + + $$ + \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen + = \sum_{n=0}^{\infty} \mu(A_n). + $$ +{% enddefinition %} + +{% definition Measure Space %} +A *measure space* is a triple $(X,\mathcal{A},\mu)$ of +a set $X$, +a σ-algebra $\mathcal{A}$ on $X$ +and a measure $\mu$ on $\mathcal{A}$. +{% enddefinition %} |